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Quantum contextual finite geometries from dessinsd’enfants.

Michel Planat, Alain Giorgetti, Frédéric Holweck, Metod Saniga

To cite this version:Michel Planat, Alain Giorgetti, Frédéric Holweck, Metod Saniga. Quantum contextual finite geome-tries from dessins d’enfants.. 2013. �hal-00873461v2�

Quantum contextual finite geometries from

dessins d’enfants

Michel Planat1, Alain Giorgetti2, Frederic Holweck3 and

Metod Saniga4

1 Institut FEMTO-ST/MN2S, CNRS, 32 Avenue de l’Observatoire, 25044 Besancon,

France (michel.planat@femto-st.fr)

2 Institut FEMTO-ST/DISC, Universite de Franche-Comte, 16 route de Gray,

F-25030 Besancon, France (alain.giorgetti@femto-st.fr)

3 Laboratoire IRTES/M3M, Universite de Technologie de Belfort-Montbeliard,

F-90010 Belfort, France (frederic.holweck@utbm.fr)

4 Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranska Lomnica,

Slovak Republic (msaniga@astro.sk)

Abstract.

We point out an explicit connection between graphs drawn on compact Riemann

surfaces defined over the field Q of algebraic numbers — so-called Grothendieck’s

dessins d’enfants — and a wealth of distinguished point-line configurations. These

include simplices, cross-polytopes, several notable projective configurations, a number

of multipartite graphs and some ‘exotic’ geometries. Among them, remarkably, we find

not only those underlying Mermin’s magic square and magic pentagram, but also those

related to the geometry of two- and three-qubit Pauli groups. Of particular interest is

the occurrence of all the three types of slim generalized quadrangles, namely GQ(2,1),

GQ(2,2) and GQ(2,4), and a couple of closely related graphs, namely the Schlafli and

Clebsch ones. These findings seem to indicate that dessins d’enfants may provide us

with a new powerful tool for gaining deeper insight into the nature of finite-dimensional

Hilbert spaces and their associated groups, with a special emphasis on contextuality.

PACS numbers: 03.65.Fd, 03.67.-a, 02.20.-a, 02.10.Ox

MSC codes: 11G32, 81P13 ,81P45, 14H57, 81Q35

1. Introduction

If one draws a (connected) graph — a particular set of vertices and edges — on a smooth

surface, then such graph inherits extra local/combinatorial and global/topological

features from the surface. If the latter is, for example, a (compact) complex

one-dimensional surface — a Riemann surface, then the combinatorics of edges is

encapsulated by a two-generator permutation group and the Riemann surface happens

to be definable over the field Q of algebraic numbers. This observation is central to the

concept of dessins d’enfants (or child’s drawings) as advocated by Grothendieck in his

Esquisse d’un programme (made available in 1984 following his Long March written in

2

1981) in the following words: In the form in which Belyi states it, his result essentially

says that every algebraic curve defined over a number field can be obtained as a covering

of the projective line ramified only over the points 0, 1 and ∞. The result seems to

have remained more or less unobserved. Yet it appears to me to have considerable

importance. To me, its essential message is that there is a profound identity between

the combinatorics of finite maps on the one hand, and the geometry of algebraic curves

defined over number fields on the other. This deep result, together with the algebraic

interpretation of maps, opens the door into a new, unexplored world - within reach of

all, who pass by without seeing it [1, Vol. 1], [2].

Our aim is to show that Grothendieck’s dessins d’enfants (see, e. g., [3, 4] as well as

[5]) have, as envisaged in [6], great potential to become a proper language for a deeper

understanding of various types of sets of Hermitian operators/observables that appear in

finite-dimensional quantum mechanical settings and for furnishing a natural explanation

why eigenvalues of these operators are regarded as the only available tracks in associated

measurements. The main justification of our aim is provided by the fact that dessins lead

very naturally to already-discovered finite geometries underlying quantum contextuality

(like the grid, GQ(2, 1), behind Mermin’s magic square and/or an ovoid of PG(3, 2)

behind Mermin’s magic pentagram) and also to those underlying commutation relations

between elements of the two-qubit Pauli group (the generalized quadrangle of order two,

GQ(2, 2), its geometric hyperplanes and their complements, see, e. g., [7]).

The paper is organized as follows. Section 2 gathers some basic knowledge

about dessins d’enfants, their permutation group and topology, their isomorphism

with conjugacy classes of the cartographic group C+2 , as well as about associated

Belyi functions. Section 3 focuses on a rather elementary application of our ideas by

interpreting Bell’s theorem about non-locality in terms of the geometry as simple as a

square/quadrangle, which is found to be generated by four distinct dessins defined over

the field Q[√2]. Section 4, the core one, starts with a complete catalog of all connected

geometries induced by dessins having up to 12 edges and a sketch of important ones

with more edges. In the subsequent subsections, we analyze in detail the non-trivial

cases by selecting, whenever possible, a dessin of genus zero and having the smallest

number of faces. As in most cases the edges of dessins dealt with admit labeling by

two- or three-qubit observables, on our way we not only encounter already recognized

quantum-relevant finite geometries like the Fano plane, the grid GQ(2, 1), the Petersen

graph, the Desargues configuration and the generalized quadrangle GQ(2, 2), but find

a bunch of novel ones, some already surmised from different contexts, starting from

the Pappus 93-configuration and the Hesse (94, 123)-configuration (aka the affine plane

AG(2, 3)), going through the Reye (124, 163)-configuration and a 3 × 3 × 3-grid (aka a

Segre variety of type S1,1,1) to arrive at the generalized quadrangle GQ(2, 4) — and its

close siblings, the Clebsch and Schlafli graphs – known to play a role in the context of

the black-hole–qubit correspondence [8]. Section 5 is reserved for concluding remarks.

3

2. Dessins d’enfants and the Belyi theorem

Dessins d’enfants and their symmetry groups

A map is a graph drawn on a surface — a smooth compact orientable variety of

dimension two — such that its vertices are points, its edges are non-intersecting arcs

connecting the vertices, and the connected components of its complement, called faces,

are homeomorphic to open disks of R2. They may exist multiple edges as well as loops,

but the graph has to be connected. Denoting the number of vertices, edges and faces

by V , E and F , respectively, the genus g of the map follows from Euler’s formula

V − E + F = 2 − 2g. A map can be generalized to a bicolored map. The latter is

a map whose vertices are colored in black and white in such a way that the adjacent

vertices have always the opposite color; the corresponding segments are the edges of the

bicolored map. The Euler characteristic now reads 2− 2g = B +W + F − n, where B,

W and n stands for the number of black vertices, the number of white vertices and the

number of edges, respectively. Given a bicolored map with n edges labeled from 1 to n,

one can associate with it a permutation group P = 〈α, β〉 on the set of labels such that

a cycle of α (resp. β) contains the labels of the edges incident to a black vertex (resp.

white vertex), taken, say, in the clockwise direction around this vertex; thus, there are

as many cycles in α (resp. β) as there are black (resp. white vertices), and the degree of

a vertex is equal to the length of the corresponding cycle. An analogous cycle structure

for the faces follows from the permutation γ satisfying αβγ = 1.

Bicolored maps (allowed to have any valency for their vertices) are in one-

to-one correspondence with hypermaps [13]. They correspond to the conjugacy

classes of subgroups of finite index of the free group on two generators H+2 =

〈ρ0, ρ1, ρ2|ρ0ρ1ρ2 = 1〉. The number of hypermaps with n edges is given by the OEIS

sequence number A057005: see http://oeis.org/A057005. Hypermaps are, of course,

allowed to have any valency for their vertices.

We consider bicolored maps where the valency of white vertices is ≤ 2. They

correspond to hypermaps of the so-called pre-clean type. As already observed by

Grothendieck himself, who called them dessins d’enfant (child’s drawings) [2]–[12], these

bicolored maps on connected oriented surfaces are unique in the sense that they are in

one-to-one correspondence with conjugacy classes of subgroups of finite index of the

triangle group, also called cartographic group

C+2 =

⟨

ρ0, ρ1, ρ2|ρ21 = ρ0ρ1ρ2 = 1⟩

. (1)

The existence of associated dessins of prescribed properties can thus be straightforwardly

checked from a systematic enumeration of conjugacy classes of C+2 ; with the increasing

n the number of such dessins grows quite rapidly

1, 3, 3, 10, 15, 56, 131, 482, 1551, 5916, 22171, 90033, 370199, · · ·A dessin D can be ascribed a signature s = (B,W, F, g) and the full information

about it can be recovered from the structure of the generators of its permutation group

4

P (also named the passport in [4, 11]) in the form [Cα, Cβ, Cγ], where the entry Ci,

i ∈ {α, β, γ} has factors lni

i , with li denoting the length of the cycle and ni the number

of cycles of length li.

Belyi’s theorem

Given f(x), a rational function of the complex variable x, a critical point of f is a root

of its derivative and a critical value of f is the value of f at the critical point. Let

us define a so-called Belyi function corresponding to a dessin D as a rational function

f(x) of degree n embedded into the Riemann sphere C in such a way that (i) the black

vertices are the roots of the equation f(x) = 0 with the multiplicity of each root being

equal to the degree of the corresponding (black) vertex, (ii) the white vertices are the

roots of the equation f(x) = 1 with the multiplicity of each root being equal to the

degree of the corresponding (white) vertex, (iii) the bicolored map is the preimage of

the segment [0, 1], that is D = f−1([0, 1]), (iv) there exists a single pole of f(x), i. e. a

root of the equation f(x) = ∞, at each face, the multiplicity of the pole being equal to

the degree of the face, and, finally, (v) besides 0, 1 and ∞, there are no other critical

values of f .

It can be shown that to every D there corresponds a Belyi function f(x) and that

this function is, up to a linear fractional transformation of the variable x, unique. It

is, however, a highly non-trivial task to find/calculate the Belyi function for a general

dessin.

Finite geometries from dessins d’enfants

An issue of central importance for us is the fact that one can associate with a dessin Da point-line incidence geometry, GD, in the following way. A point of GD corresponds

to an edge of D. Given a dessin D, we want its permutation group P to preserve the

collineation of the geometry GD. Thus given S a subgroup of P which stabilizes a pair

of points, we define the point-line relation on GD such that two points will be adjacent

if their stabilizer is isomorphic to S. This construction allows to assign finite geometries

GDi to a dessin D, i = 1, · · · , m with m being the number of non-isomorphic subgroups

S of P that stabilize a pair of elements ‡. As a slight digression we mention that,

presumably, this action of the group P of a dessin D on the associated geometry GD

is intricately linked with the properties of the absolute Galois group Γ = Gal(Q/Q),

which is the group of automorphisms of the field Q of algebraic numbers. Although

Γ is known to act faithfully on D [2, 12], its action on GD must be rather non-trivial

because the map from D to GD is non injective. Further work is necessary along this

line of thoughts to clarify the issue.

‡ Our definition follows an example of the action on the Fano plane of a permutation group of order

|PSL(2, 7| = 168 associated with a tree-like dessin (of the relevant cycle structure) given in [2, (a), vol.

2, p. 17 and p. 50].

5

Using a computer program, we have been able to completely catalogize incidence

geometries associated with dessins featuring up to 12 edges, and also found several

dessins of higher rank that produce distinguished geometries. The results of our

calculations are succinctly summarized in Tables 1 and 2. The subsequent sections

provide a detailed account of a variety of dessins computed, their corresponding point-

line incidence geometries, and, what is perhaps most important, how these relate to

the physics of quantum observables of multiple-qubit Pauli groups and related quantum

paradoxes. In other words, we shall give a more exhaustive and rigorous elaboration of

the ideas first outlined in a short essay-like treatise [6].

3. The square geometry of Bell’s theorem and the corresponding dessins

In a theory in which parameters are added to quantum mechanics to determine the results

of individual measurements, without changing the statistical predictions, there must be

a mechanism whereby the setting of one measuring device can influence the reading of

another instrument, however remote [9].

The square geometry of Bell’s theorem

Suppose we have four observables σi, i = 1, 2, 3, 4, taking values in {−1, 1}, of whichBob can measure (σ1, σ3) and Alice (σ2, σ4). The Bell-CHSH approach to quantum

contextuality/non-locality consists of defining the number

C = σ2(σ1 + σ3) + σ4(σ3 − σ1) = ±2

and observing the (so-called Bell-CHSH) inequality [17, p. 164]

| 〈σ1σ2〉+ 〈σ2σ3〉+ 〈σ3σ4〉 − 〈σ4σ1〉 | ≤ 2,

where 〈〉 here means that we are taking averages over many experiments. This inequality

holds for any dichotomic random variables σi that are governed by a joint probability

distribution. Bell’s theorem states that the aforementioned inequality is violated if one

considers quantum observables with dichotomic eigenvalues. An illustrative example is

the following set of two-qubit observables

σ1 = IX, σ2 = XI, σ3 = IZ, σ4 = ZI, (2)

where X , Y and Z are the ordinary Pauli spin matrices and where, e. g., IX is a short-

hand for I ⊗X (used also in the sequel). We find that

C2 = 4 ∗ I + [σ1, σ3][σ2, σ4] = 4

1 . . 1

. 1 1 .

. 1 1 .

1 . . 1

6

index name vertices edges triangles squares

3 2-simplex (triangle) 3 3 1 0

4 3-simplex (tethahedron) 4 6 4 0

square/quadrangle 4 4 0 1

5 4-simplex (5-cell) 5 10 10 0

6 5-simplex 6 15 20 0

3-orthoplex (octahedron) 6 12 8 3

bipartite graph K(3, 3) 6 9 0 9

7 6-simplex 7 21 35 0

Fano plane (73) 7 21 7 0

8 7-simplex 8 28 56 0

4-orthoplex (16-cell) 8 24 32 6

completed cube K(4, 4) 8 16 0 36

stellated octahedron 8 12 8 0

9 8-simplex 9 36 84 0

Hesse (94123) 9 36 12 0

extended-Pappus 9 27 27 27

Pappus (93) 9 27 9 27

(3× 3)-grid 9 18 6 9

10 9-simplex 10 45 120 0

5-orthoplex 10 40 80 10

bipartite graph K(5, 5) 10 25 0 100

Mermin’s pentagram 10 30 30 15

Petersen graph 10 15 10 0

Desargues (103) 10 30 10 15

11 10-simplex 11 55 165 0

12 11-simplex 12 66 220 0

6-orthoplex 12 60 160 15

bipartite graph K(6, 6) 12 36 0 255

threepartite graph K(4, 4, 4) 12 48 0 108

fourpartite graph K(3, 3, 3, 3) 12 54 0 54

Reye (124163) 12 24 16 6

Table 1. A catalog of connected point-line incidence geometries induced by dessins

d’enfants of small index ≤ 12. For each geometry, when represented by its collinearity

graph, we list the number of points, edges (line-segments joining two points), triangles

and squares it contains. Here, A-simplices should be regarded as trivial because their

dessins are star-like and associated Belyi functions are of a simple form f(x) = xA,

where A is the multiplicity of the singular point at x = 0 [4].

has eigenvalues 0 and 8, both with multiplicity 2 (1 ≡ −1). Taking the norm of

the bounded linear operator A as ||A|| = sup(||Aψ||/||ψ||), ψ ∈ H (H being the

7

index name vertices edges triangles squares

15 Cremona-Richmond (153) (alias GQ(2, 2)) 15 45 15 90

16 Clebsch graph CG: sp(101, 25,−210) 16 80 0 60

Shrikhande graph SG: sp(61, 26,−29) 16 48 32 12

18 toroidal deltahedron(?): sp(81, 09,−44, 24) 18 72 48 306

20 sp(010,−81, 81,−24, 24) 20 80 0 740

21 Kneser graph KG(7,2): sp(101, 36,−214) 21 105 35 630

L(IG(7, 3, 1)) : sp(41,−28, (1±√2)6) 21 42 14 0

22 IG(11, 5, 2): sp(±51,±√310) 22 55 0 55

27 (3× 3× 3)-cube: sp(61, 36, 012,−38) 27 81 27 81

GQ(2, 4): sp(101, 120,−56) 27 135 45 1080

Schlafli graph SHG: sp(161, 46,−220) 27 216 720 270

sp(161, 116,−28,−82) 27 216 504 3024

sp(161, 42, 112,−28,−54) 27 216 612 1674

Table 2. A few (non-trivial) connected point-line incidence geometries induced by

dessins d’enfants of index greater than 12. The spectra of (collinearity) graphs,

denoted as sp(· · · , ab, · · ·) where an eigenvalue a is of multiplicity b, are also displayed.

IG(ν, k, λ) means the incidence graph of a symmetric 2 − (ν, k, λ) design and L(· · ·)stands for the line graph.

corresponding Hilbert space), one arrives at the maximal violation of the Bell-CHSH

inequality [17, p. 174], namely ||C|| = 2√2.

The point-line incidence geometry associated with our four observables is one of

the simplest, that of a square – Fig.1a; each observable is represented by a point and

two points are joined by a segment if the corresponding observables commute. It is

worth mentioning here that there are altogether 90 distinct squares among two-qubit

observables and as many as 30240 when three-qubit labeling is employed [15], each

yielding a maximal violation of the Bell-CHSH inequality.

Dessins d’enfants for the square and their Belyi functions

As it is depicted in Fig. 1, the geometry of square can be generated by four different

dessins, b1, · · · , b4, associated with permutations groups P isomorphic to the dihedral

group D4 of order 8.

The first dessin (b1) has the signature s = (B,W, F, g) = (3, 2, 1, 0) and the

symmetry group P = 〈(2, 3), (1, 2)(3, 4)〉 whose cycle structure reads [2112, 22, 41], i.e.

one black vertex is of degree two, two black vertices have degree one, the two white

vertices have degree two and the face has degree four. The corresponding Belyi

function reads f(x) = x2(2 − x2). Its critical points are x ∈ {−1, 1, 0} and the

corresponding critical values are {1, 1, 0}. The preimage of the value 0 (the solutions

of the equation f(x) = 0) corresponds to the black vertices of the dessin positioned

8

Figure 1. A simple observable proof of Bell’s theorem is embodied in the geometry

of a (properly labeled) square (a) and four associated dessins d’enfants, b1 to b4. For

each dessin an explicit labeling of its edges in terms of the four two-qubit observables

is given. The (real-valued) coordinates of black and white vertices stem from the

corresponding Belyi functions as explained in the main text.

at x ∈ {−√2,√2, 0} and the preimage of the value 1 (the solutions of the equation

f(x) = 1) corresponds to the white vertices at x = ±1. The second dessin (b2) has

s = (2, 3, 1, 0), P = 〈(1, 2)(3, 4), (2, 3)〉 with [22, 2112, 41], and the Belyi function of the

form f(x) = (x2 − 1)2. The third dessin (b3) is characterized by s = (1, 2, 3, 0) and

its P = 〈(1, 2, 4, 3), (1, 2)(3, 4)〉 has the cycle structure [41, 22, 2112]. The Belyi function

may be written as f(x) = (x−1)4

4x(x−2). As f ′(x) = (x−1)3(x2

−2x−1)2(x−2)2x2 , its critical points lie at

x = 1 (where f(1) = 0) and at x = 1±√2 (where f(1 ±

√2) = 1). Finally, the fourth

dessin (b4) has P = 〈(1, 2, 4, 3), (2, 3)〉, the signature s = (1, 3, 2, 0) and cycle structure

[41, 2112, 22]. The Belyi function reads f(x) = (x−1)4

16x2 ; hence, f ′(x) = (x−1)3(x+1)8x3 . The

critical points are at x = −1 (with critical value 1) and x = 1 (with critical value 0), the

preimage of 0 is the black vertex at x = 1 and the preimage of 1 consists of the white

vertices at x ∈ {−1, 3±√8}.

Summing up, one of the simplest observable proofs of Bell’s theorem is found to

rely on the geometry of a square and four distinct dessins associated with it. Although

we still do not know how these dessins are related to each other, it is quite intriguing

to see that all critical points live in the extension field Q(√2) ∈ Q of the rational field

Q. Hence, a better understanding of the properties of the group of automorphisms of

9

this field (which is itself a subgroup of the absolute Galois group Γ) may lead to fresh

insights into the nature of this important theorem of quantum physics.

4. A wealth of other notable point-line geometries relevant to contextuality

It is also appealing to see the failure of the EPR reality criterion emerge quite directly

from the one crucial difference between the elements of reality (which, being ordinary

numbers, necessarily commute) and the precisely corresponding quantum mechanical

observables (which sometimes anti-commute) [14, (a)].

4.1. Two geometries of index six: the octahedron and the bipartite graph K(3, 3)

As the geometries of index five are only trivial simplices (see Table 1), we have to move

to index six in order to encounter non-trivial guys, namely the octahedron and the

bipartite graph of type K(3, 3).

The octahedron can be labeled by three-qubit observables, and one such labeling

is given in Fig. 2a. The figure also illustrates one of the associated dessins (b), whose

Belyi function of is f(x) = 2732x2(2 − x2)2. The function has critical points at x = 0

and x = ±√2 (these being also the preimage of 0), and the white vertices of the dessin

correspond to x = ±√

2/3 and x = ±√

8/3 (the preimage of 1).

A remarkable property of the graph K(3, 3) is that it lives in the generalized

quadrangle GQ(2, 2), disguised there as a generalized quadrangle of type GQ(1, 2) [18].

And since GQ(2, 2) was found to mimic the commutation relations between elements of

the two-qubit Pauli group [7], K(3, 3) thus naturally lends itself, like the above-discussed

square, to a labeling in terms of two-qubit observables — as, for example, depicted in

Fig. 3a. One of the associated dessins (Fig. 3b) possesses the Belyi function of the form

f(x) = ax4(x − 1)2, where a = 36

24= 729

16. Since f ′(x) = ax3(x − 1)(3x − 2) the critical

points are at x = 2/3, x = 0 and x = 1. The black vertices of the dessin correspond

to x = 0 and x = 1; out of its five white vertices three answer to real-valued variable,

namely x = −13, x = 2

3and x ≈ 1.118 (the latter being denoted as x1 in Fig. 3b), and

the remaining two — denoted as x2 and x3 in the figure in question — have imaginary,

complex-conjugate one: x ≈ 0.36 exp (±iφ), with φ ≈ 99.4◦.

4.2. The Fano plane (everywhere)

The only non-trivial geometry of index seven is the projective plane of order two, the

Fano plane (Fig. 4a). This plane plays a very prominent role in finite-dimensional

quantum mechanics, being, for example, intricately related — through the properties of

the split Cayley hexagon of order two [18] — to the structure of the three-qubit Pauli

group [19]. A quick computer search for all permutation subgroups of C+2 isomorphic

to the group PSL(2, 7), the automorphism group of the Fano plane, shows that this

plane can be recovered from 10 distinct dessins. One choice is depicted schematically

in Fig. 4b; it corresponds to passport 8 (the fourth dessin) in the catalog of Betrema &

10

Figure 2. The octahedron with vertices labeled by three-qubit observables (a) and

an associated dessin (b).

Figure 3. The bipartite graph K(3, 3) with one of its two-qubit labelings (a) and its

generating dessin (b).

11

Figure 4. The Fano plane portrayed in its most frequent rendering (a) and one of its

ten stabilizing dessins (b).

Zvonkin [16]. The corresponding permutation group is P = 〈(2, 7, 6, 5)(3, 4), (1, 2)(3, 5)〉and the Belyi function reads f(x) =

√8x4(x − 1)2(x − a), with a = −1

4(1 + i

√7); its

critical points are located at x = 0 and x = 1 (yielding critical value 0) and at x = a

(yielding 1).

4.3. The 16-cell, stellated octahedron and the completed cube

When moving to index eight, we encounter an appealing 16-cell on the one hand, and

the remarkably “twinned” stellated octahedron and completed cube on the other hand.

The 16-cell (Fig. 5a) arises from a “straight-line” dessin with the signature (5, 4, 1, 0)

and the permutation group isomorphic to D8. Its Belyi function is the fourth order of

the map x→ x2−2, that is f(x) = 8x(x2−2)(x4−4x2+2), with critical points located

at x− 0, x = ±√2 and x = ±

√

2±√2.

The dessin with signature s = (2, 6, 2, 0) , illustrated in Fig. 6c, has the permutation

group P = 〈(1, 2, 4, 3)(5, 7, 6, 8), (2, 5)(3, 7)〉, which is isomorphic to Z32⋊Z2 and endowed

with the cycle structure of the form [42, 2214, 42]. The stabilizer of a pair of its edges

is either the group Z2, leading to the geometry of a stellated octahedron (Fig. 6a), or

the trivial single element group Z1, in which case we get the geometry of a (triangle

free) completed cube, i. e. the ordinary cube where pairs of opposite points are joined

(Fig. 6b). The latter configuration also appears in a recent paper [20, pp. 33–34 as well

as Conjecture 6.1] as an 8-face Kepler-Poinsot quadrangulation of the torus. Note that,

in addition the 6 faces shared with the ordinary cube, the completed cube contains also

8 non-planar faces of which four are self-intersecting. The completed cube can also be

12

Figure 5. The 16-cell (a) and an associated dessin (b).

Figure 6. The stellated octahedron (a), the completed cube (b) and their common

stabilizing dessin (c).

13

viewed as the bipartite graph K(4, 4). The Belyi funcion of the dessin has the form

f(x) = K(x− 1)4(x− a)4

x3, a =

8√10− 37

27, K ≈ −0.4082,

from where we find the positions of “critical” white vertices to be x ≈ 0.0566± 0.506i,

with the other four white vertices being located at x = −1.069, x = −0.162 and

x = 1.634± 0.6109i.

4.4. Geometries of index nine: grid (Mermin’s square), Pappus and Hesse

In the realm of index nine we meet, in addition to our old friend, a 3 × 3-grid (alias

generalized quadrangle GQ(2, 1)), also other distinguished finite geometries like the

Pappus 93-configurations and the Hesse 94123-configuration (aka the affine plane of

order three, AG(2, 3)).

As already mentioned, the grid lives (as a geometric hyperplane) in GQ(2, 2) and

underlies a Mermin “magic” square array of observables furnishing a simple two-qubit

proof of the Kochen-Specker theorem [6, 21]. A Mermin square built around Bell’s

square of Fig. 1a is shown in Fig. 7a. It needs a genus one dessin, with signature

(2, 5, 2, 1), to be recovered, as shown in Fig. 7b. The corresponding permutation group is

P = 〈(1, 2, 4, 8, 7, 3)(5, 9, 6), (2, 5)(3, 6)(4, 7)(8, 9)〉 ∼= Z23 ⋊Z2

2, having the cycle structure

[6131, 2411, 6131]. This dessin lies on a Riemann surface that is a torus (not a sphere C),

being thus represented by an elliptic curve. The topic is far more advanced and we shall

not pursue it in this paper (see, e. g., [12] for details). The stabilizer of a pair of edges

of the dessin is either the group Z2, yielding Mermin’s square M1 shown in Fig 7a, or

the group Z1, giving rise to a different square M2 from the maximum sets of mutually

non-collinear pairs of points of M1. The union of M1 and M2 is nothing but the Hesse

configuration.

The Hesse configuration (Fig. 8a), of its own, can be obtained from a genus-zero

dessin shown in Fig. 8b (also reproduced in Fig. 3b of [6]). This configuration was already

noticed to be of importance in the derivation of a Kochen-Specker inequality in [22].

The Pappus configuration, illustrated in Fig. 9a, comprises three copies of the

already discussed K(3, 3)-configuration (Fig. 3a); the three copies are represented by the

point-sets {1, 3, 5, 6, 7, 8}, {2, 3, 4, 5, 8, 9} and {1, 2, 4, 6, 7, 9}, which pairwise overlap in

distinct triples of points. A dessin d’enfant for the Pappus configuration is exhibited in

Fig. 9b. It is important to point out here a well-known fact that the Pappus configuration

is obtained from the Hesse one by removing three mutually skew lines from it (for

example, the three lines that are represented in Fig. 8a by three concentric circles).

4.5. Realm of index ten: Mermin’s pentagram, Petersen and Desargues

Apart from the two plexes (see Table 1), the only connected configurations generated

by 10-edge dessins are Mermin’s pentagram, the Petersen graph, the Desargues

configuration and the bipartite graph K(5, 5).

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Figure 7. A 3 × 3 grid with points labeled by two-qubit observables (aka a Mermin

magic square) (a) and a stabilizing dessin drawn on a torus (b).

Figure 8. The Hesse configuration (a) and an associated genus-zero dessin (b).

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Figure 9. The Pappus configuration (a) and a stabilizing dessin (b).

The dessin sketched in Fig. 10c, having s = (4, 6, 2, 0) and the alternating group A5

with cycle structure [3211, 2412, 52], induces either the geometry of Mermin’s pentagram

(Fig. 10a) or that of the Petersen graph (Fig. 10b) according as the group stabilizing pairs

of its edges is isomorphic to Z1 or Z2, respectively. A particular three-qubit realization

leading to a proof of the Kochen-Specker theorem is explicitly shown (see, e. g., [15, 21]

for more details on importance of these geometries in quantum theory).

The dessin depicted in Fig. 11b also gives rise to a couple of geometries, one being

again the Petersen graph (with the stabilizer group of a pair of edges isomorphic to Z22)

and the other being (Fig. 11a) the famous Desargues 103 configuration (with the group

Z2). The labeling is compatible with that in Fig. 10, which means that the Desargues

configuration represents another way of encoding a three-qubit proof of contextuality;

in particular, a line of Mermin’s pentagram corresponds to a complete graph K4 within

the Desargues configuration as well as to a maximum set of mutually disjoint vertices

in the Petersen graph.

4.6. The Cremona-Richmond 153-configuration, alias GQ(2, 2), or W(3, 2)

We now come to a perhaps most exciting, and encouraging as well, finding that there

exists a dessin generating the configuration of a central importance for any quantum

physical reasoning involving two-qubit observables, namely the configuration (illustrated

in Fig. 12a) known as 153 Cremona-Richmond configuration, or the generalized

quadrangle of order two, GQ(2, 2), or the symplectic polar space of rank two and order

two, W(3, 2). That this configuration is indeed one of corner-stones of finite dimensional

quantum mechanics is also illustrated by the fact that many of the already discussed

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Figure 10. The Mermin pentagram (a), the Petersen graph (b) and their generating

dessin (c).

Figure 11. The Desargues configuration (a) and its generating dessin (b).

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Figure 12. The Cremona-Richmond 153-configuration (a) with its points labeled by

the elements of the two-qubit Pauli group and its stabilizing dessin (b).

geometries, in particular the K(3, 3) graph, the 3 × 3 grid, the Pappus and Desargues

configurations and the Petersen graph, are intricately tied to its structure, as explained

in detail in [7],[23]–[25]. The associated dessin (Fig. 12b) is of signature (5, 9, 3, 0) and

its permutation group has the cycle structure [61322111, 2613, 6231]. Unfortunately, the

complexity of this dessin is already so high that with our current computer power we

have not been able to find the corresponding Belyi function. Finding this function thus

remains one important challenge of our dessin d’enfant programme.

4.7. The generalized quadrangle GQ(2, 4), the Schlafli graph and the Clebsch graph

As already mentioned, the generalized quadrangle of type GQ(2, 4) is a prominent

finite geometry in the context of the so-called black-hole–qubit correspondence [8], as

it completely underlies the E6-symmetric entropy formula describing black holes and

black strings in D = 5. We were thus very pleased to find a dessin that leads to the

collinearity graphs of this geometry, and to its complement – the famous Schlafli graph

– as well (Fig. 13). Moreover, GQ(2 ,4) is also notable by the fact it contains (altogether

27) copies of the Clebsch graph, each such copy being the complement of a geometric

hyperplane of particular type. And since the Clebsch graph is also a dessin-generated

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Figure 13. The dessin for both the collinearity graph of the generalized quadrangle

of type GQ(2, 4) and its complement, the Schlafli graph.

one (see Table 2), we will not be surprised if our dessin’s formalism is also found of

relevance for getting conceptual insights into the still-mysterious formal link between

stringy black hole entropy formulas and properties of multi-qubits (for a recent review,

see [27]).

5. Conclusion

We have demonstrated, substantially boosting the spirit of [6], that Grothendieck’s

dessins d’enfant (“child’s drawings”) — that is graphs where at each vertex is given a

cyclic ordering of the edges meeting it and each vertex is also assigned one of two colors,

conventionally black and white, with the two ends of every edge being colored differently

— and their associated permutation groups/Belyi functions give rise to a wealth of finite

geometries relevant for quantum physics. We have made a complete catalog of these

geometries for dessins featuring up to 12 edges, highlighted distinguished geometries for

some higher-index dessins, and briefly elaborated on quantum physical meaning for each

non-trivial geometry encountered. We are astonished to see that a majority of dessin-

generated geometries have already been found to have a firm footing in finite-dimensional

quantum mechanical setting, like the K(3, 3) and Petersen graphs, the Fano plane, the

3 × 3 grid (Mermin’s square), the Desargues configuration, Mermin’s pentagram and

the generalized quadrangles GQ(2, 2) and GQ(2, 4). We have also found a wealth of

geometries, among them the Hesse 94123-configuration, the Reye 124163-configuration

[26], the 3 × 3 × 3-grid, the Kneser graph KG(7,2) and many others, that still await

19

their time to enter the game. Our findings may well be pointing out that properties of

dessins, as well as the Galois group G = Gal(Q/Q) acting on them, may be vital for

getting deeper insights into foundational aspects of quantum mechanics. To this end in

view, we aim to expand in a systematic way our catalog of finite geometries generated

by higher-index dessins in order to reveal finer traits of the quantum pattern outlined

above.

Acknowledgments

This work started while two of the authors (M.P. and M. S.) were fellows of the

“Research in Pairs” Program of the Mathematisches Forschungsinstitut Oberwolfach

(Oberwolfach, Germany), in the period from 24 February to 16 March, 2013. It was

also supported in part by the VEGA Grant Agency, grant No. 2/0003/13.

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